Your thesis statement would, of course, need to be directly related to the argument you wanted to make about identity. However, given the brevity and tight focus of the story, you could make a strong argument built around one of these thesis statements:

Louise Mallard's identity is so intertwined with her position as a wife that changing that definition kills her.

And/or:

The importance of freedom to personal identity can be seen in how Mrs. Mallard dies when she learns her husband is still alive.